The link between 2n , and the ApÃƒ ryâ€™s numbers an , bn is given by taking y = n in (22) and
e
2n
t = n in (23):
2 2
n n+k
n

2n (n) = = bn ;
k k
k=0

n
1 1
2n (n) + 2n (n) = an :
k3 2
k=1

ApÃƒ ry was the Ã¿rst to prove irrationality of (3). He only used recurrence relation between the an
e
and bn . We end the proof of irrationality of (3) with the error term for the PadÃƒ approximation.
e
Let us recall equality (21),
n
1 1 1
(3) = + 3
k3 2 n
k=1
M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250
e 245